56.4.15 problem 15

Internal problem ID [8904]
Book : Own collection of miscellaneous problems
Section : section 4.0
Problem number : 15
Date solved : Sunday, March 30, 2025 at 01:52:56 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x -2\right ) y^{\prime \prime }+\frac {y^{\prime }}{x}+\left (x +1\right ) y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 2 \end{align*}

✓ Maple. Time used: 0.031 (sec). Leaf size: 46

Order:=6; 
ode:=(x-2)*diff(diff(y(x),x),x)+1/x*diff(y(x),x)+(1+x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=2);
 

\[ y = c_1 \sqrt {x -2}\, \left (1-\frac {23}{12} \left (x -2\right )+\frac {127}{160} \left (x -2\right )^{2}+\frac {1621}{40320} \left (x -2\right )^{3}-\frac {426599}{5806080} \left (x -2\right )^{4}+\frac {4670443}{425779200} \left (x -2\right )^{5}+\operatorname {O}\left (\left (x -2\right )^{6}\right )\right )+c_2 \left (1-6 \left (x -2\right )+\frac {31}{6} \left (x -2\right )^{2}-\frac {37}{45} \left (x -2\right )^{3}-\frac {299}{840} \left (x -2\right )^{4}+\frac {6743}{56700} \left (x -2\right )^{5}+\operatorname {O}\left (\left (x -2\right )^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 105

ode=(x-2)*D[y[x],{x,2}] + 1/x*D[y[x],x] + (x+1)*y[x] ==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,2,5}]
 

\[ y(x)\to c_1 \left (\frac {4670443 (x-2)^5}{425779200}-\frac {426599 (x-2)^4}{5806080}+\frac {1621 (x-2)^3}{40320}+\frac {127}{160} (x-2)^2-\frac {23 (x-2)}{12}+1\right ) \sqrt {x-2}+c_2 \left (\frac {6743 (x-2)^5}{56700}-\frac {299}{840} (x-2)^4-\frac {37}{45} (x-2)^3+\frac {31}{6} (x-2)^2-6 (x-2)+1\right ) \]

✓ Sympy. Time used: 1.214 (sec). Leaf size: 82

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((x - 2)*Derivative(y(x), (x, 2)) + (x + 1)*y(x) + Derivative(y(x), x)/x,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=2,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (- 6 x + \frac {73 \left (x - 2\right )^{5}}{945} - \frac {19 \left (x - 2\right )^{4}}{210} - \frac {22 \left (x - 2\right )^{3}}{15} + \frac {17 \left (x - 2\right )^{2}}{3} + 13\right ) + C_{1} \sqrt {x - 2} \left (- 2 x - \frac {5 \left (x - 2\right )^{4}}{126} - \frac {2 \left (x - 2\right )^{3}}{21} + \left (x - 2\right )^{2} + 5\right ) + O\left (x^{6}\right ) \]